\(W^{4, p}\) solution to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations

DOI10.1016/j.jde.2015.11.013zbMath1336.35147arXiv1506.06727OpenAlexW2963557879MaRDI QIDQ2634266

Nam Q. Le

Publication date: 8 February 2016

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1506.06727

zbMATH Keywords

Legendre transformsecond boundary value problemAbreu's equationlinearized Monge-Ampère equationaffine mean curvature equation

Mathematics Subject Classification ID

Smoothness and regularity of solutions to PDEs (35B65) Boundary value problems for higher-order elliptic equations (35J40) Affine differential geometry (53A15) Quasilinear elliptic equations with mean curvature operator (35J93) Monge-Ampère equations (35J96)

Related Items

Solvability of a class of singular fourth order equations of Monge-Ampère type ⋮ Global \(W^{2,p}\) regularity on the linearized Monge-Ampère equation with \(\text{VMO}\) Type coefficients ⋮ Harnack inequality for the fractional nonlocal linearized Monge-Ampère equation ⋮ On certain degenerate and singular elliptic PDEs I: nondivergence form operators with unbounded drifts and applications to subelliptic equations ⋮ Regularity of Hamiltonian stationary equations in symplectic manifolds ⋮ Interior estimates for the Monge-Ampère type fourth order equations ⋮ \(W^{1,p}_{\varphi}\)-estimates for Green's functions of the linearized Monge-Ampère operator ⋮ Approximation of variational problems with a convexity constraint by PDEs of Abreu type ⋮ Interior \(C^{1,\alpha}\) regularity for the linearized Monge-Ampère equation with VMO type coefficients ⋮ On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity ⋮ Boundary Harnack inequality for the linearized Monge-Ampère equations and applications

Cites Work

• Unnamed Item
• Some minimization problems in the class of convex functions with prescribed determinant
• The Abreu equation with degenerated boundary conditions
• Monge-Ampère functionals and the second boundary value problem
• Extremal metrics on toric surfaces: a continuity method
• Constant scalar curvature metrics on toric surfaces
• The Bernstein problem for affine maximal hypersurfaces
• Elliptic partial differential equations of second order
• Scalar curvature and stability of toric varieties.
• Boundary regularity for the Monge-Ampère and affine maximal surface equations
• Periodic solutions of Abreu's equation
• Interior estimates for the \(n\)-dimensional Abreu's equation
• Interior estimates for solutions of Abreu equation
• The First Boundary Value Problem for Abreu’s Equation
• Properties of the solutions of the linearized Monge-Ampere equation
• 𝐾-stability on toric manifolds
• KÄHLER GEOMETRY OF TORIC VARIETIES AND EXTREMAL METRICS
• The affine Plateau problem
• Pointwise 𝐶^{2,𝛼} estimates at the boundary for the Monge-Ampère equation
• Global Second Derivative Estimates for the Second Boundary Value Problem of the Prescribed Affine Mean Curvature and Abreu's Equations
• A generalised Joyce construction for a family of nonlinear partial differential equations